# Completeness in the theory of types

@article{Henkin1950CompletenessIT, title={Completeness in the theory of types}, author={Leon Henkin}, journal={Journal of Symbolic Logic}, year={1950}, volume={15}, pages={81 - 91} }

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system. For the functional calculus of second order, in which predicate variables may be bound, a very different kind of result is known: no matter what (recursive) set of axioms are chosen, the system will contain a…

## 797 Citations

### Some forms of completeness

- MathematicsJournal of Symbolic Logic
- 1962

If a theory doesn't have any individual constants it must necessarily admit as terms expressions constructed by means of operators those constructed by mean of operators such as the epsilon or iota operators.

### A nonconstructive proof of Gentzen’s Hauptsatz for second order predicate logic

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- 1966

Takeuti [3] showed that the consistency of analysis (i.e. second order number theory) is finitistically implied by the Hauptsatz for second order logic» i.e. by the proposition that every theorem of…

### Second-order languages and mathematical practice

- PhilosophyJournal of Symbolic Logic
- 1985

There are no straightforward analogues to the Lowenheim-Skolem theorems for second-order languages and logic, and some controversy in recent years as to whether “second-order logic” should be considered a part of logic, but this boundary issue does not concern me directly.

### Resolution in type theory

- PhilosophyJournal of Symbolic Logic
- 1971

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very…

### Systems of transfinite types involving λ-conversion

- MathematicsJournal of Symbolic Logic
- 1953

The problem of constructing formal systems involving transfinite types was briefly suggested many times in the literature. Actually few attempts have been made to develop such systems. In particular,…

### On Languages Which are Based on Non-Standard Arithmetic

- Philosophy, MathematicsNagoya Mathematical Journal
- 1963

The natural numbers play a part in the formulation of logical syntax inasmuch as they are used to count the symbols in a sentence, or the sentences in a proof, etc. In the present paper, we shall…

### Semantic Characterizations of Number Theories

- Philosophy
- 1992

The first-order predicate calculus is complete for its intended semantics, by Godel’s Completeness Theorem. Type Theory, though not complete for its intended semantics, is complete for the more…

### Forcing and the Omitting Type Theorem , institutionally

- Mathematics
- 2011

In the context of proliferation of many logical systems in the area of mathematical logic and computer science, we present a generalization of forcing in institution-independent model theory which is…

### Course Notes in Typed Lambda Calculus

- Philosophy
- 1998

Both have a common origin in logic. The first notion of types seem to come from Frege: he revealed the conceptual difference between objects and predicates and considered the hierarchy built on these…

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